The HAB (or some other stretchy wrap, there are many brands, many of which are cheaper than the HAB) is a long piece of material (in fact you can make your own) which you wrap around yourself, then put hte baby in. Lasts up till around 8kg (some use it longer, but it will generally start to feel less secure and saggy with heavier babies).
It's very very secure for baby and very comfortable to use. the weight is very evenly distributed across your back and hips.

The close carrier probably cannot be used for as long since the weight is not so broadly distributed - more on your shoulders. Also, because it's somewhat structured - with the rings to adjust - it won't fit everyone (a wrap will fit anyone).

not sure about the brace. certainly some slings/carriers work better with braces than others....